Aim of the article and pre-requisites

By the end of this hands-on tutorial you should be familiar with:

The minimum required knowledge to benefit from the article includes being familiar with:

A familiar background in systematic reviews

You are leading a research project that involves conducting a systematic review with multiple interventions for a specific health condition and patient population. You and your team have spent a considerable amount of time and energy to set-up a protocol of high standards and you have registered the protocol to PROSPERO register. You have exhausted the (re)sources to search, retrieve and select the relevant trials that will represent your target population as close as possible.

The data extraction is coming to an end, and you are looking forward to uncovering the best interventions for the investigated condition. Everybody is motivated, and overwhelmed, yet quite stressed to keep up with the deadlines of the project. You pour some coffee in your favorite mug and you start the analysis. You notice that many eligible trials report to have missing participants. How are you going to proceed with the analysis?

The good, the bad, and the ugly

Consider the following fictional small trial in Figure 1 for illustration: 20 participants have been randomised to two interventions. The outcome is binary, such as the symptoms have improved or deteriorated after receiving the randomised intervention. Some participants have completed the trial (called completers), and others left the trial prematurely for several reasons (called missing). We are interested in drawing conclusions using the whole randomised sample.

Figure 1: A fictional small trial of two interventions

Let’s be honest; without access to individual patient data, there are only a few methods to address missing participants in meta-analysis:

  1. exclude missing participants under the assumption that their outcome is missing for reasons unrelated to the planning and conduct of the trial (missing completely at random, MCAR) or their missing outcome is attributed to their baseline characteristics (missing at random, MAR),
  2. impute missing participants under the assumption that they would have experienced improvement or deterioration of their symptoms, had they never left the trial (missing not at random, MNAR) and take this assumption for granted,
  3. model the missingness mechanism (i.e., M(C)AR or MNAR) assuming a probability distribution that describes the association between the missing participants and
    the completers concerning their outcomes (pattern-mixture model).

Exclusion

As you may have guessed, exclusion and imputation are suboptimal methods to handle missing participants. Exclusion is illustrated in Figure 2: missing participants in both interventions are removed from the dataset before conducting the analysis. This methods is inappropriate for the goals of our study, where we aim to draw conclusions for the whole randomised sample, as it reduces the sample to those participants who remained in the trial. Consequently, the power to detect a statistically significant association is also reduced. Moreover, if the missingness mechanism is not M(C)AR, the risk of estimating a biased intervention effect (e.g. odds ratio) is imminent.

Figure 2: Excluding missing participants from both interventions

Imputation

Figure 3 illustrates the imputation: all missing participants in the first intervention have been assumed to have experienced improvement of their symptoms (and thus, they left the trial prematurely), whilst all missing participants in the second intervention have been assumed to have experienced deterioration of their symptoms (and thus, they left the trial prematurely). Imputation implies adding the number of missing participants to the outcome dictated by the corresponding scenario: to the number of events (improvement) in the first intervention, and to the number of non-events (deterioration) in the second intervention. This method maintains the randomised sample but takes the missingness scenario for granted. Without following the missing participants to record their outcome, we cannot know the true missingness mechanism. Hence, any assumptions we make about the missingness mechanism should naturally propagate in the estimation of the intervention effect by increasing its standard error. Imputation rears its ugly head by ‘treating’ the missing participants as observed, increasing the risk to estimate a biased intervention effect and draw spurious conclusions regarding the statistical significance.

Figure 3: Imputing missing participants making different scenarios in each intervention

It is needless to say that meta-analysis inherits the limitations of exclusion and imputation observed at trial-level via the inclusion of studies with missing participants, giving you a real headache. Remember, you do not have individual patient data to make use of different cool methods for missing participants. Hence, the quality of the meta-analysis results will strongly dependent, among others, on your good guess about the missingness mechanism and how you incorporated this good guess into the results having access only to a handful of methods with some being suboptimal. We have good news on that; just, keep reading.

Pattern-mixture model

The last member of the squad of methods for missing participants in meta-analysis is the pattern-mixture model. The pattern-mixture model circumvents the limitations of exclusion and imputation elegantly: it adjusts, rather than fixes, the average outcome (here, the risk of event) to the assumed missingness mechanism, and propagates this assumption to the standard error of the estimated intervention effect, while maintaining the randomised sample. It also allows the assumption about the missing mechanism to differ between the interventions or across the trials. Therefore, the pattern-mixture model offers a flexible modelling framework to handle missing participants in meta-analysis properly. If exclusion, imputation and pattern-mixture model were characters from the epic Western spaghetti film, The Good, the Bad and the Ugly, the correspondence of who would be who is quite obvious.

Figure 4 illustrates two normal distributions of the informative missingness odds ratio (IMOR), a central parameter in the pattern-mixture model for a binary outcome. IMOR resembles the odds ratio for two interventions: it compares the odds of an event in missing participants with the odds of an event in completers. IMOR equal to 1 coincides with the MAR mechanism (i.e., no difference in the compared groups), and IMOR different from 1 implies the MNAR mechanism (i.e., the compared groups differ). As an illustration, we have considered a different IMOR distribution for each intervention: the first intervention is assumed to have on average twice the odds of an event in completers than that in missing participants (the mode), with other scenarios being less and less likely to occur as we deviate from the mode. On the contrary, the second intervention is assumed to have on average twice the odds of an event in missing participants than that in completers (the mode), again, with other scenarios being less and less likely to occur as we deviate from the mode. An important technical detail is that the normal distributions presented in Figure 4 are implied for the IMOR in the logarithmic scale, also commonly considered when estimating the odds ratio.

Figure 4: The IMOR parameter under different scenarios for each intervention

The versatility of the rnmamod R package

Originally, we developed the rnmamod R package as a response to facilitate handling aggregate missing participants in the absence of relevant functions in the available R packages for pairwise and network meta-analysis. Since then, one thing led to another and we ended up expanding our goals to set-up a comprehensive suite of functions to perform and visualise Bayesian pairwise and network meta-analysis. Proper handling of missing participants remains fundamental and is implemented using the pattern-mixture model in all models of the package. The rest of the tutorial walks you down to the core functions of the package to perform Bayesian network meta-analysis and obtain a compact, yet comprehensive visual summarisation of a large volume of results.

Let’s get started

You may install and load the package directly from CRAN by running the code below:

install.packages("rnmamod")
library(rnmamod)

However, we recommend installing the development version to experience the latest advances in the package:

install.packages("devtools")
devtools::install_github("LoukiaSpin/rnmamod")

We will use the dataset of Baker et al. (2009) that includes 21 trials comparing seven pharmacological interventions with each other and placebo in chronic obstructive pulmonary disease (COPD) patients. We are interested in whether the patients experienced COPD exacerbation after receiving the randomised intervention. Run head(nma.baker2009) to see the first six trials, or nma.baker2009 to glance at the whole dataset. The dataset has the one-trial-per-row format, which is the typical format encountered in models written in the BUGS language.

head(nma.baker2009)
#>                 study t1 t2 t3 t4 r1 r2 r3 r4 m1 m2 m3 m4  n1  n2 n3 n4
#> Llewellyn-Jones, 1996  1  4 NA NA  3  0 NA NA  1  0 NA NA   8   8 NA NA
#>        Paggiaro, 1998  1  4 NA NA 51 45 NA NA 27 19 NA NA 139 142 NA NA
#>          Mahler, 1999  1  7 NA NA 47 28 NA NA 23  9 NA NA 143 135 NA NA
#>        Casaburi, 2000  1  8 NA NA 41 45 NA NA 18 12 NA NA 191 279 NA NA
#>       van Noord, 2000  1  7 NA NA 18 11 NA NA  8  7 NA NA  50  47 NA NA
#>         Rennard, 2001  1  7 NA NA 41 38 NA NA 29 22 NA NA 135 132 NA NA

Network plot and evidence description

Let’s create the network plot using the netplot function (type netplot to learn more). If the plot looks familiar, it is because netplot currently calls the nma.networkplot function from the pcnetmeta R package:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# Create the network plot and tables
netplot(data = nma.baker2009, 
        drug_names = interv_names, 
        text.cex = 1.5)

The netplot function gives further insights into the network evidence by prints on the console the followings three tables:

  • A description of the network evidence

  • A summary of the trials, randomised sample, and outcome data for each intervention

  • A summary of the trials, randomised sample, and outcome data for each observed comparison in the network

Mapping the evidence on missing participants

The nma.baker2009 dataset was selected for the substantial number of missing participants in most trials, interventions and comparisons. We use the heatmap_missing_dataset to have a look at how missing participants (in per cent) are distributed in the dataset. The presence of missing participants is quite intense; only a few trials have interventions with zero or low missingness. Type ?heatmap_missing_dataset to find out more about this function.

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# Create the heatmap of % missing participants for each trial and arm
heatmap_missing_dataset(data = nma.baker2009, 
                        trial_names = nma.baker2009$study,
                        drug_names = abbr_interv_names)

Use the heatmap_missing_network function to view the median and range of missing participants (in per cent) for each intervention (main diagonal) and comparison (lower off-diagonal). We can immediately spot the comparison with the lowest median of missing participants: tiotropium versus formoterol. Of note, we have used colours that are friendly to people with colour vision deficiency. Find out more about this function by typing ?heatmap_missing_network.

# Abbreviated interventions (in the order they appear in the dataset)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# Create the heatmap of % missing participants for each intervention and observed comparison
heatmap_missing_network(data = nma.baker2009, 
                        drug_names = abbr_interv_names)

The fine architecture of the rnmamod package

Now you will be introduced to the backbone function of the rnmamod architecture: the run_model. In short, this is the function to conduct Bayesian network meta-analysis while accounting for the missing participants (if extracted) via the pattern-mixture model. As you will discover in the next chapters of the tutorial, most functions in the package cannot work without the run_model function. At the same time, the run_model function is not much of use as a stand-alone function: it merely returns the results for a bulk of model parameters (and they are a lot). The magic begins once the run_model function is fed into the remaining functions of the package. Then you get to run all necessary analyses that accompany network meta-analysis and obtain a plethora of graphs to help you understand the results and outline your conclusions, while wearing your critical hat.

Conduct a random-effects network meta-analysis

Without further ado, let’s run a Bayesian random-effects (model = "RE") network meta-analysis while, at the same time, addressing the missing participants reported in each intervention arm of every trial. The effect measure of interest is the odds ratio (measure = "OR"). We consider a half-normal prior distribution on the between-trial standard deviation with scale parameter equal to 1 (heter_prior = list("halfnormal", 0, 1)). We will estimate one IMOR parameter for each intervention in the network (assumption = "IDE-ARM") under the MAR assumption on average (mean_misspar = c(0, 0)) with a variance equal to 1 (var_misspar = 1). The investigated outcome is harmful (D = 0). Make sure to assign a name to the function to be used as an object in other functions of the model. Note that run_model and all other model functions of the package run in JAGS.

## Run random-effects network meta-analysis
res <- run_model(data = nma.baker2009,
                 measure = "OR",
                 model = "RE",
                 assumption = "IDE-ARM",
                 heter_prior = list("halfnormal", 0, 1),
                 mean_misspar = c(0, 0), 
                 var_misspar = 1,
                 D = 0,
                 n_chains = 3,
                 n_iter = 10000,
                 n_burnin = 1000,
                 n_thin = 1)

On the console you see will two sets of progress bars: the first one shows the progress of the simulation. Once the model completes the iterations for all chains, a second progress bar appears on the console that shows the progress of the model update for n_iter = 10000 iterations until convergence:

Specifying carefully the arguments of the run_model function is essential for the subsequent analyses to yield sensible results. This requires thorough consideration of the proper effect measure (measure), the meta-analysis model (model), the structure of the missingness parameter (assumption), the prior distribution for the heterogeneity parameter (heter_prior), the missingness mechanism (mean_misspar and var_misspar), the direction of the outcome (D), and the necessary tuning parameters to run Markov chain Monte Carlo simulations using JAGS (n_chains, n_iter, n_burning, and n_thin). Refer to the documentation of the function by typing ?run_model.

Now we can use res in the arguments of the other model functions to conduct the subsequent analyses, and visualise the results. The journey has just begun.

Markov chain Monte Carlo diagnostics

We will use the homonym function, mcmc_diagnostic, to find out how our model fared in terms of convergence.

# Display diagnostics for Bayesian methods
mcmc_diagnostics(net = res,
                 par = c("EM", "tau"))

The function prints on the console a message on good diagnostic practices (in red), followed by the names of panel plots and tables on several monitored parameters.

Here, we illustrate a panel of bar plots on the log odds ratio for all possible comparisons in the network (x-axis):

Then mcmc_diagnostics opens a browser with a panel of diagnostic plots (trace, density, and autocorrelation) for the monitored parameters specified in the par argument (here, the log odds ratio for all possible pairwise comparisons, "EM", and the common between-trial standard deviation, "tau"). Type res$jagsfit to access all monitored parameters from the run_model function. The HTML file is created thanks to the mcmcplot function of the R-package mcmcplots.

The league tables

Estimation

Use the league_heatmap function to create the popular league table by running the following code:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# The heatmap league table for estimation 
league_heatmap(full1 = res, 
               drug_names1 = interv_names)

The table is read as row versus column. For instance, budesonide plus formoterol (row) versus budesidone (column) is associated with the (posterior mean) odds ratio of 1.28 and a 95% credible interval (0.47, 3.60). The interventions are sorted in decreasing order by their posterior mean SUCRA value shown in the main diagonal. The larger the treatment effect, the darker the colour shade. The outcome is harmful, hence, blue favours the first intervention of the comparison, and red favours the second intervention. The league_heatmap function can also display the treatment effects from two outcomes. For more details, type ?league_heatmap.

Prediction

The league_heatmap_pred function has the same arguments with the league_heatmap function, but it displays the predictions for all possible pairwise comparisons in the network:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# The The heatmap league table for prediction 
league_heatmap_pred(full1 = res, 
                    drug_names1 = interv_names)

Intervention ranking and relevant measures

The rankosucra_plot function creates a plot for each intervention that combines two popular hierarchy measures: the rankogram and SUCRA plot:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# The rankogram-SUCRA-plot 'amalgamation'
rankosucra_plot(full1 = res, 
                drug_names1 = interv_names)

The interventions are sorted from best to worst based on their SUCRA value (the number on the top left corner of each plot). The rankosucra_plot function can also display the hierarchy results from two outcomes; just, type ?rankosucra_plot for more details.

Pairwise versus network meta-analysis

We have created the functions run_series_meta and series_meta_plot to help us explore the implications in the estimated treatment effects and between-trial standard deviation of assuming consistency and common heterogeneity parameter as compared to performing pairwise meta-analysis, separately, for each observed comparison in the network.

First, we apply the run_series_meta function that conducts separate pairwise meta-analyses for all observed comparisons in the network:

# Run separate pairwise meta-analyses
meta <- run_series_meta(full = res)

Here, the function inherits the arguments n_chains, n_iter, n_burning, and n_thin from the run_model function via res; however, we can also specify these arguments anew. Like with run_model, the console prints the progress of simulations together with the number of models to run: there are 15 observed comparisons in our network; hence, the console prints this message (in red), followed by a progress bar for each analysis alongside the model update until convergence:

Next, insert meta in the function series_meta_plot to experience the visualisation toolkit for this analysis:

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# End-user-ready results for a series of pairwise meta-analyses
series_meta_plot(full = res,
                 meta = meta,
                 drug_names = abbr_interv_names)

This function returns the same results in two formats: (i) a table that is printed on the console, and (ii) a panel of forest plots.

Network meta-analysis led to more precise results than pairwise meta-analysis for all observed comparisons (plot A)). The benefits of conducting network meta-analysis for this example are also evident for the heterogeneity parameter, which was estimated at a reasonable level and with greater precision than any pairwise meta-analysis (plot B)): blue vertical lines refer to the posterior median and 95% credible interval of the common heterogeneity parameter in network meta-analysis.

By the way, you can export the table to an xlsx file at your working directory by inserting the argument save_xls = TRUE in the series_meta_plot function.

Comparison of network with pairwise meta-analysis results should never be aimed at drawing conclusions about the plausibility of consistency assumption! There are tools specifically devised to evaluate the consistency assumption. In the next part of the tutorial, you will be introduced to the functions of the package that offer local and global evaluation of the consistency assumption.

Consistency evaluation

The node-splitting approach (local evaluation)